Hamiltonian systems, symplectic flows, classical integrable systems

**3**

votes

**1**answer

101 views

### Duistermaat-Heckman integral formula on compact manifold with boundary

Let a compact Lie group $G$ acts on a closed symplectic manifold $(M,\omega)$. If the action is Hamiltonian with $\mu$ the moment map, then the integral $$\int_M e^{i\mu (X)+\omega}$$ is equal to the ...

**2**

votes

**0**answers

134 views

### Ricci flat metric on pair (X,D)

Let $(X,\omega)$ be a Calabi-Yau variety and $D$ be a simple normal crossing divisor on $X$ with conic singularities with cone angle $2\pi\theta$, $0<\theta<1$ such that $K_X+D>0$, then is ...

**5**

votes

**0**answers

71 views

### Uniqueness of Birkhoff Normal Form and KAM theory for Symplectomorphims

I am starting to work with Hamiltonian Dynamics and I have been taking a look at some of the basic stuff in KAM theory. I have posted part of this question at MSE but as I did not get any response I ...

**3**

votes

**0**answers

58 views

### Delzant polytopes and combinatorial types

At first, let us see the following matheoverflow question,
About a Delzant polytope. (In particular dodecahedron)
The question is whether (combinatorial) regular dodecahedron can be realized as a ...

**2**

votes

**0**answers

175 views

### Integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of varieties of general type is a rational numbers?

It is known that the integrations of Ricci curvature of the Weil-Petersson metric on the moduli space of Calabi-Yau varieties is a rational numbers
My question is about the moduli space of ...

**3**

votes

**0**answers

90 views

### Fibers of torus equivariant moment maps

Given a closed (possibly singular) projective variety $V$ with a symplectic structure and a torus action, there is a moment map
$\mu: V \rightarrow Lie(T)^*$. Note that the dimension of $T$ could be ...

**1**

vote

**0**answers

88 views

### What is this 2-form on a Lagrangian torus fibration?

Suppose we are given a regular $2n$-dimensional Lagrangian fibration $\pi : (M,\omega) \to B$ with connected, compact fibers. Then it is well-known (Arnold-Liouville) that each fibre is a Lagrangian ...

**8**

votes

**2**answers

629 views

### What is the obstruction to the existence of a global Kahler potential?

It is a well-know fact that if $(X,\omega)$ is Kahler then about every point $x \in X$ there exists a neighbourhood $U$ and a function $K \in C^{\infty}(U,\mathbb{R})$ such that $\omega|_U = i\partial ...

**2**

votes

**0**answers

398 views

### Weil Petersson metric on moduli space of Calabi Yau manifolds

Let $f:(X,D)\to Y$ be a holomorphic fibre space where $D$ is divisor with conic singularities and let fibres $(X_s,D_s)$ are log Calabi-Yau pair .i.e $K_X+D$ is nummerically trivial, then we have ...

**2**

votes

**0**answers

106 views

### holomorphic curves in almost toric fibration and their relation to tropical curves

My goal is to get better understanding how the projection of holomorphic curves converge to tropical disks.
We are given an almost toric fibration $X\rightarrow B$ with special Lagrangian fibers with ...

**2**

votes

**2**answers

370 views

### $dd^\mathbb{C}$-lemma on pair $(X,D)$

Let $X$ be a Kähler manifold with a simple normal crossing divisor $D$, i.e., pair $(X,D)$. Let $\omega$ and $\omega'$ be two Kähler forms in the same Kähler class then have we $dd^\mathbb{C}$-lemma ...

**2**

votes

**0**answers

122 views

### Flow on invariant Lagrangian tori

The most concrete version of the question is :
A (necessarily) invariant Lagrangian torus $L$ on the unit cotangent of a Riemannian metric on the two-torus carries a periodic orbit with period $T$. ...

**3**

votes

**1**answer

132 views

### Condition on moment polytope for a toric manifold to be Fano

Suppose $M$ is a symplectic toric manifold. This means there is a compact torus
$T$ that has a Hamiltonian action on $M$, with moment map $\mu:M \to \mathfrak t^*$, and $\dim(M)=2\dim(T)$. Can one ...

**6**

votes

**0**answers

166 views

### Handle attachment in symplectic category

It is known that for an exact symplectic manifold $(M,\omega_M)$ with a convex boundary $(\partial M,\theta_M)$, where $d\theta_M=\omega_M$ (usually called a Liouville domain), one can attachment to ...

**11**

votes

**1**answer

274 views

### Pseudo-holomorphic curves in the six-sphere

Equip $S^6$ with the almost complex structure coming from the cross product on $\mathbb R^7$ (i.e. the product on the pure imaginary octonions). What is known about the psudo-holomorphic curves in ...

**1**

vote

**0**answers

43 views

### Is the preimage of coisotropic submanifold coisotropic?

Let $M$ and $N$ be smooth manifolds with Poisson-structures $\{ \cdot , \cdot\}|_M$ and $\{\cdot , \cdot \}|_N$ We call $\phi: M \to N$. a Poisson-map, if the pullback of $\phi$ is compatible with the ...

**2**

votes

**0**answers

118 views

### From symplectic manifold to Hilbert spaces [closed]

What could be a mathematical model of such physical wish? I'm looking for something sending a symplectic manifold $(M,\omega)$ to a Hilbert space $H_{M}$ with the following properties:
1- We should ...

**13**

votes

**0**answers

313 views

### What is the Hochschild cohomology of the Fukaya-Seidel category?

Let $(Y, \omega)$ be a compact symplectic manifold and let $Fuk(X,\omega)$ be its Fukaya category. The Hochschild cohomology of this category should be given by $HH^\bullet(Fuk(Y,\omega))=H^\bullet(Y, ...

**7**

votes

**0**answers

278 views

### Question about theorem in Arnold's book on action-angles variables

I have a question about the action-angle theorem on p. 283 in Arnold's textbook on classical mechanics.(I added the link to this book in the last part of this question)
If you don't have the book or ...

**2**

votes

**0**answers

108 views

### Is there an algorithm to compute the intersection of tautological classes on the moduli space of genus one curves?

Let $\overline{M}_{1,1}(\mathbb{P}^2, d) $ be the moduli space of degree
$d$ genus one curves on $\mathbb{P}^2$ with one marked point. Let
$L\longrightarrow \overline{M}_{1,1}(\mathbb{P}^2, d) $ ...

**0**

votes

**1**answer

101 views

### Unique Equivariant Symplectic Structure for the Full Flag Manifold of $SU(3)$?

I was looking at the following interesting question about the number of equivariant almost complex structures on the full flag manifold of $SU(3)$, and I began to wonder how many equivariant ...

**3**

votes

**1**answer

153 views

### Local symplectomorphisms become global ones?

It is widely known that a local diffeomorphism is not necessarily a global diffeomosphism and so on.
Now, I stumbled over the question whether in some particular cases, as I will describe below, ...

**2**

votes

**2**answers

207 views

### Blowup and Delzant Polytope

Can someone please explain why blow up in a Symplectic toric manifold corresponds to chopping off a corner in the Delzant polytope?

**5**

votes

**1**answer

190 views

### Does pseudo-holomorphic *submanifolds* satisfy unique continuation?

Let $f,g:(D^2,j_\mathrm{std})\to(B^{2n}(1),J)$ be two pseudo-holomorphic maps. The following unique continuation result is well-known (it may be proved using either Aronszajn's Lemma or the Carleman ...

**12**

votes

**0**answers

186 views

### Is tightness decidable?

Given a contact structure on a three-manifold, is there an algorithm to decide whether or not it tight?
For concreteness' sake, let's agree to represent the given contact three-manifold via an open ...

**2**

votes

**1**answer

172 views

### How can we define constant scalar curvature Kahler or cscK on pair $(X,D)$

A Kahler metric $\omega$
with cone singularities along divisor
$D$
with cone angle $2\pi\beta$
is said to be
of
constant scalar curvature Kahler
or
cscK
if its scalar curvature $S(\omega)$, which is ...

**2**

votes

**0**answers

132 views

### Poisson Manifold Structures on Even Dimensional Spheres

The $2n$-sphere, for $n=1,2,3$, possess a (non-trivial) Poisson manifold structure. Is this still true for $n > 3$? Describing the spheres as homogeneous spaces $SO(n)/S(n-1)$, are there Poisson ...

**0**

votes

**1**answer

259 views

### Steps in paper on sympl. geometry unclear

I am currently reading a paper on symplectic geometry: Periodic orbits for Hamiltonian systems in cotangent bundles by Christopher Golé.
It deals with the question how the stability properties of a ...

**1**

vote

**0**answers

79 views

### Poisson algebra automorphisms of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. Let $V=\mathcal{C}^\infty(M)$ be the Lie algebra of smooth real valued functions. Suppose $f:\rightarrow V$ be an Lie-algebra isomorphism (an algebra ...

**3**

votes

**1**answer

139 views

### Symplectic geometry and stability of orbits

I am looking for a theorem (read it once, but forgot about it and would now like to find a reference with proof):
In symplectic geometry (at least for some particular subset of $\mathbb{R}^2$), there ...

**3**

votes

**1**answer

130 views

### Action generated by geodesic flow is hamiltonian

I'm trying to understand why a certain action of a Lie Group is hamiltonian.
Let $(M,g)$ be a geodesically complete Riemannian manifold.
Then there exists a canonical one-form on the cotangentbundle $...

**4**

votes

**1**answer

187 views

### Check symplectomorphism property on infinitesimal generators

I stumbled over the following question:
First, let me give the basic definition of a symplectic group action:
Let $(M, \omega)$ be a symplectic manifold and $G$ a Lie group. A smooth action $\Phi:G \...

**0**

votes

**0**answers

151 views

### Is the complex structure on a del-Pezzo surface a regular complex structure?

Let $(X, \omega, J)$ be a compact symplectic manifold with an almost complex structure. Fix some homology class $\beta \in H_2(X, \mathbb{Z})$. An almost
complex structure $J$ is said to be $\textit{...

**3**

votes

**2**answers

248 views

### Legendrian knot in 3-sphere

We are given a Legendrian knot, fixed up to Legendrian isotopy, in $(S^3,\xi)$ ($\xi$ is the standard contact structure). Does it necessarily bound a symplectic surface in $(B^4,\omega)$ (again $\...

**2**

votes

**1**answer

159 views

### Hamiltonian flow local diffeomorphism?

I am currently reading Arnold's proof of the Darboux theorem in his book on classical mechanics and fail to understand some point.
The background
So he wants to show that any symplectic form is ...

**1**

vote

**0**answers

148 views

### Symplectic spectrum

I have a question about a step in the proof of the following theorem from symplectic geometry.
The theorem is: Given any ellipsoid $E:=\{ w \in \mathbb{R}^{2n}; \sum_{i,j =1}^{2n} a_{ij}w_iw_j \le 1\}...

**1**

vote

**0**answers

73 views

### Limiting behaviour of the symplectic form

It is well known that the coadjoint orbits of the Heisenberg group (with a suitable choice of coordinate system) are the planes $z=c\ne 0$ parallel to the $xy$-plane, and the points in the $xy$-plane. ...

**5**

votes

**1**answer

95 views

### multiplicity free chain of Lie subgroups gives a completely integrable system on symplectic manifold

Assume we have a symplectic manifold $N$ and a Hamiltonian action of a Lie group $G$, such that the algebra of all $G$-invariant functions on $N$ is commutative under Poisson bracket.
Furthermore, ...

**1**

vote

**0**answers

87 views

### An algebraics Hamiltonian vector field with a finite number of periodic orbits(2)

Is there a polynomial Hamiltonian $H:\mathbb{R}^{4}\to \mathbb{R}$ such that the number of nontrivial periodic orbits of the corresponding Hamiltonian vector field $X_{H}$ is finite but different ...

**1**

vote

**0**answers

286 views

### Proof of Arnold-Liouville theorem in classical mechanics [closed]

I am currently reading Arnold's book "Mathematical Methods of classical mechanics" on page 278 and I don't see through his arguments there at a point.
Especially, I am talking about the part that ...

**0**

votes

**1**answer

167 views

### Non Hamiltonian vector field

Let $\Phi: G \times M \rightarrow M$ be a group action on a symplectic manifold $M$ and $G$ be a Lie group.
Furthermore, $x$ is a solution of the Hamilton equation $\dot{x}(t) = X_H(x(t))$ and for a ...

**0**

votes

**0**answers

69 views

### Characterisation of vector fields solution to a simple equation

This question is complementary to another question I asked on math.stackexchange. I believe it is more subtle than it seems - it will become clearer when I provide more context - and probably hides ...

**0**

votes

**1**answer

160 views

### When is the normal neighbourhood of the boundary of the moduli space of cuvres parametrized by exactly one branch?

Let $X$ be a compact complex manifold and $\beta \in H_2(X, \mathbb{Z}) $
a fixed homology class that is $\textit{decomposable}$. Let
$$ \overline{\mathcal{M}}_{0,n}(X, \beta) $$
denote the stable ...

**0**

votes

**0**answers

178 views

### How does one define Moduli spaces in Symplectic Geometry and naively interpret higher genus GW Invariants?

This is a very basic question about the definition of Moduli space of maps.
My reason for asking this question is because I haven't actually seen this
definition explicitly given anywhere, and hence ...

**2**

votes

**0**answers

417 views

### Vafa's semi-Ricci flat metric

Cumrun Vafa with Greene-Shapere-Yau introduced semi-Ricci flat metric here
B. Greene, A. Shapere, C. Vafa, and S.-T. Yau. Stringy cosmic strings and
noncompact Calabi-Yau manifolds. Nuclear Physics B,...

**0**

votes

**0**answers

133 views

### What is the symplectic manifold whose Delzant polytope is a trapezoid?

What is the symplectic form on the manifold whose associated Delzant polytope is a trapezoid? I am trying to find it by using the Marsden–Weinstein theorem, but I have been unable to do so. If ...

**6**

votes

**1**answer

544 views

### Symplectic reversing diffeomorphisms on a compact symplectic manifold

I Ask this question in MSE and I received interesting comments and ideas. I repeat the question here for more discussion:
Let $(M,\omega)$ be a compact symplectic manifold.
Is there a ...

**2**

votes

**1**answer

212 views

### Boundary conditions and the relationship between Hamiltonian and Lagrangian Floer theories

I am looking for a full account of the relationship between the various versions of Floer theory on a symplectic manifold $M$. If we take the usual Floer equation (Hamiltonian version)
\begin{equation*...

**0**

votes

**1**answer

191 views

### Moment maps and flat degenerations of toric varieties

We have a flat family of projective varieties with a torus $T$ action, over $\mathbb{A}^1$.
How do the moment map images of the fibers change when we pass from the generic fiber to the special fiber ...

**20**

votes

**2**answers

764 views

### Are symplectic methods used in (classical) Economics?

The tl;dr question is this: are economists using coordinate-free formulations in studying theory?
Borrowing from classical mechanics, the framework I have in mind for classical economics--involving ...